I'm interested in numerically modelling viscoelastic fluids in a 2D channel geometry, driven by a pressure drop. The method consists of solving coupled constitutive equation: (a) Stokes force balance (for incompressible flow) and (b) a polymeric model equation, e.g., Oldroyd-B or similar, which contributes an additional viscoelastic stress into (a). Eqn (a) is subject to no-slip and no-permeation boundary conditions (BCs). The pressure drop is in the x-direction, and the wall-normal direction is y. The details of the viscoelastic component is not necessary for the question - it is mentioned (i) to provide context for my problem, and (ii) as it means that solutions beyond simple Pouseille flow can be found.
Intuitively it seems as though solutions with a net y-velocity component should not allowed as this implies that the sample as a whole is moving vertically up or down in the channel, violating the no-permeation BCs. Vortex solutions where positive vy on one side of the vortex is cancelled out by negative vy on the other are clearly OK.
However I am struggling to show this analytically. Does anyone have any advice for how to show this if true, or can they provide a counter-example if not.
Many thanks!
After some discussion with the office, we have came up with the following solution (special thanks to David Hoyle). Basically you integrate the continuity equation along an x-slice at arbitrary y and the use the BCs to show that this must be zero.
I believe no. For Stokes flow with a Newtonian fluid it is easy to show. You difficulty probably arises from the non-linearities of the Oldroyd-B model. Maybe there is some asymptotic limit that you can investigate at small Deborah number ?
Depending on the geometry of the material and it's interaction with the fluid, yes it is possible. But in the wall normal direction the flow will die very soon.
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